A particle of mass $\text{'}m\text{'}$ is rigidly attached at $\text{'}A\text{'}$ to a ring of mass $\text{'}3 \mathrm{m}\text{'}$ and radius $\text{'}r\text{'}.$ The system is released from rest and rolls without sliding. The angular acceleration of the ring just after

A uniform smooth rod (mass $m$ and length $\mathcal{l}$ ) placed on a smooth horizontal floor is hit by a particle (mass $m$) moving on the floor, at a distance $\frac{\mathcal{l}}{4}$ from one end elastically ($e=1$). The distance travelled by the centre of the rod after the collision, when it has completed three revolutions, will be

Two identical uniform solid spherical balls $A$ and $B$ of mass $m$ each are placed on a the fixed wedge as shown in figure. Ball $B$ is kept at rest and it is released just before two balls collides. Ball $A$ rolls down without slipping on inclined plane and collide elastically with ball $B.$The kinetic energy of ball $A$ just after the collision with ball $B$ is

A disc of radius $r=0.1 \mathrm{m}$ is rolled from a point $A$ on a track as shown in the figure. The part $AB$ of the track is a semicircle of radius $R$ in a vertical plane. The disc rolls without sliding and leaves contact with the track at its highest point $B.$ Flying through the air it strikes the ground at point $C.$ The velocity of the centre of mass of the disc makes an angle of $30°$ below the horizontal at the time of striking the ground. At the same instant, velocity of the topmost point $P$ of the disc is found to be $6 \mathrm{m}/\mathrm{s}.$ The value of $R$ is (Take $g=10 \mathrm{m}/{\mathrm{s}}^2$).

A uniform $rodBC$ with length $a$ is attached to a light string $AC.$ End $A$ of the string is fixed to the ceiling and the end $B$ of the rod is on a smooth horizontal surface. $B$ is exactly below point $A$ and length $AB$ is $b(a<b<2 a).$ The system is released from rest and the rod begins to slide. Find the speed of the centre of the rod when the string becomes vertical.

A solid sphere of mass $M$ and radius $R$ is pulled horizontally on a rough surface as shown in figure. Choose the incorrect alternatives.

A disc is given an initial angular velocity ${\omega }_0$ and placed on a rough horizontal surface as shown in figure. The quantities which will not depend on the coefficient of friction is/are